PhD course: Probabilistic Number Theory

Course literature:
[1] E. Kowalski. An Introduction to Probabilistic Number Theory
(Cambridge Studies in Advanced Mathematics). Cambridge University Press,
Cambridge, 2021. doi:10.1017/9781108888226

Course description.
This course is aimed at a general PhD level audience and should be
accessible and of interest to doctoral students from different
directions, including combinatorial, analytic as well as algebraic or
geometric directions. Probabilistic methods and heuristics play an
increasingly important role in many areas of mathematics with striking
applications in proofs of deterministic results. This course aims at
developing a basic tool box of probabilistic methods based on
applications within number theory. The course will start out from
Emmanuel Kowalski’s recent book [1] on this subject, but may include
additional material in form of current research papers. A background in
number theory is not required for this course as we will follow
Kowalski’s approach and keep the number theoretic input at a minimum
while focussing on the probabilistic tools.

Intended learning outcome.
• To understand and be able to apply probabilistic techniques to analyse
the asymptotic behaviour of arithmetically defined probability measures.
In other words, to gain a basic tool box of probabilistic tools.

Course content.
This course is an introduction to applications of probabilistic methods
within number theory. We will discuss a selection of the topics
presented in [1], starting out from the Erdős-Kac theorem about the
distribution of number of distinct prime factors of a typical integer of
size about N . Possible topics include the distribution of values of the
Riemann Zeta function, Chebychev bias (which concerns the question
whether there are there more primes p ≡ 3 (mod 4) than primes p ≡ 1 (mod
4)) as well as connections between exponential sums and random walks.

Course structure.
Lectures, homework, presentations by course participants.

Prerequisites. No specific prerequisites beyond what is needed to start
a PhD in mathematics.

Examination. Homework and/or presentation